An inverse eigenvalue problem for an arbitrary multiply connected bounded region: an extension to higher dimensions

VOL. 16 NO. 3 (1993) 485-492

AN INVERSE

EIGENVALUE PROBLEM FOR AN

ARBITRARY

MULTIPLY

CONNECTED

BOUNDED REGION:

AN EXTENSION

TO HIGHER

DIMENSIONS

E.M.E.ZAYED

Mathematics

Department,

FacultyofScience Zagazig University, Zagasiz,

Egypt

(Received August 30, 1991 and in rPvised form September 3, 1992)

ABSTRACT. The basic problem in this paper is that of detemnining the geometry of an

arbitrary multiply connected bounded regionin R3_{togetherwiththe mixedboundary conditions,}

from the complete knowledge of the eigenvalues

{,}=

for the negative Laplacian, using the asymptoticexpansion ofthe spectralfunctionO(t)= ezp(-

tA)

ast-,O.

KEY WORDS AND PHRASES.

Inverse problem, Laplace’s operator, eigenvalue problem and spectralfunction.

1991AMS SUBJECT CLASSIFICATION

CODES.

35K, 35P.

1. INTRODUCTION.

The underlying problem is todeduce the precise shape of a membrane from the complete 3

(0)2

knowledge of the eigenvalues

{A}o=

for the negative Laplacian

-&3=-

E in the

10x

(z],

2,

z3)

space.

Let tic_Ra be a simply connected bounded domain with a smooth bounding surface .’;.

Considerthe

Dirichlet/Neumann

problem

(A+A)u=0infl, (1.1)

Ou

u=0 or

--

0 onS,

(1.2)

where denotes differentiation alongtheinward pointing normal to$. Denoteitseigenvalues, countedaccordingtomultiplicity,by

0< <

2

-<

<

’

< asi-"

(1.3)

The problem of determining the geometry of f has been discussed by Pleijel

[4],

McKean and Singer

[3],

Waechter

[5],

Gottlieb

[1],

Hsu

[2]

and Z,yed

[6-8, 11],

using the asymptotic expansion ofthe spectralfunction

O(t)

y]

ezp(-

t,,)

ast--O.

(1.4)

j=l

Ithas been shownthat,inthecaseofDirichletboundaryconditions

(D.b.c)

o(t)-

v

ISl

n

dS+ao+O(t)

as t-0,

(1.5)

(4rt)3/2

16xt

+

12r3/2t

_S

while,inthecaseofNeumann boundaryconditions

(N.b.c.),

o(t)=

v

ISI

H

dS+ao+O(t

as t0,

(1.6)

(4rt)3/2

+

1--’-

+

12x312t

S

(

)

is themean curvatureofS, where

R

and

R

are theprincipal radii of curvature.

Furtherxnore, the constant terma0in

(1.5)

and

(1.6)

has thefollowingforms:

f

-i

sf(--

’-}V-as,

in thecaseof D.b.c.

(see

[5]),

(1.7)

a₀

s-if(--s

-1’

Vas,

inthecaseof N.b.c.

(see

[2]).

In

termsof the meancurvature Hand Gaussiancurvature N

--,

(1.7)

may be rewritten inthe forms:

{’

f(n

N)es, in thecaseofD.b.c.,

"0 $

(1.8)

f(n

m)dS, in thecaseof N.b.c.

The objectof this paper is todiscuss thefollowing moregeneral inverseproblem: Let flbe an arbitrary multiply connectedbounded region in R3 _{which issurrounded internally by simply}

connected bounded domains fl, with smooth bounding surfaces S,,i 1,2 m-I, and externally by a simply connected bounded domain _f,, with asmooth bounding surface S,.

Suppose

that the eigenvaJues

(1.3)

aregiven for the eigenvalue equation

(A3

+

,)u 0inf,

(1.9)

togetherwithoneof the followingmixedboundaryconditions:

O.__U_oon$,,

i=l, k, u=OonS,, i=k+l...m,

(1.10)

or

=0onS,, i=

,

-,=0onS,

=+1 m,

(1.11)

o _{denote differentiations along the inward pointing normals to S,,i m. Determine}

where

thegeometryofflfrom the asymptotic form of thespectralfunctionO(t) for small positivet.

Note that problem

(1.)-(1.11)

has been investigated recently by Zayed

[11]

in the special

casewhenfisanarbitrarydoublyconnected region (i.e., m

2).

2. STATEMENT OF RESULTS.

Suppose that the bounding surfaces S,(i= m) of the region fl are given hwallv

infinitely differentiable functions r"=v"(a,),n 1,2,3, ofthe parameters tr, constants, arv

ofcurvature, thefirst and second fundamental formsofSi(i m) can be written inthefollowingforms:

l’I

(, ,) "t,i)

,

+",)

,),

and

In

terms of the coefficients a,,a2i,bl,,bi the principal raxtii of curvatures for S0(i m) are

given by:

Rlt

ali/bl,and

R2i

Consequently, the mean curvatures H, and Gaussian curvatures N, of the bounding surfaces

S,(i m)aredefinedby:

Let I$i1,(i= m) be the surface areas of the bounding surfaces S, (i m) respectively. Then,the results of problem

(1.9)-(1.11)

can be summarized in thefollowingcases:

CASE1.

(N.b.c.

on _S,,i kandD.b.c.on_S,, =/

+

_m)

(4xt)3/2 i=l i=k+l

12r3/2t

i=

S,

+

{

}

+tl..")

3

.,ds,-

,,,dS,

i

,q,

+

O()gs

CASE 2. (D.b.c. onS,,i kand N.b.c. onS,,i

+

m)

In

this case, the asyInptotic expansion of O(tl as t----(I follows directly from (2.1) with the interchangesS., k.-,S., k+

With reference to formulae (1.5), (1.6) and to the articles

[1],

[2], [7], [11],

the a.symptotic expansion

(2.1)

may be interpretedasfollows:

(i)

fl is an arbitrary multiply connected bounded region in R3 _{and we have the mixed} boundaryconditions

(1.10)

or (1.11)asindicated in thespecificationsof the two respectivecases.

(if)

For thefirst fiveterms, fl is anarbitrary multiply connected bounded region in R3_of volumeV.

In

Case the boundingsurfaces S,,i 1, k areof surfaceareas E IS,l, mean curvatures

I=1

//, and Gaussian curvature N togetherwith Neumannboundary conditions, while thebounding

surfaces S,,i k

+

in are of surface areas

IS,

I, mean curvatures H, and Gaussian

I=k+l

curvature/v, togetherwith Dirichletboundaryconditions.

Weclosethis section with the following remarks:

REMARK

2.1. Onsetting/ 0in

(2.1)

withtheusual definition that iszero, weobtain

theresult of D.b.c.onS,,i m.

REMARK

2.2.

On

setting/ min

(2.1)

with the usual definition that is zero, we I=rn+l

obtainthe result of N.b.c.onSi, m.

3. FORMULATION OF THE MATHEMATICAL PROBLEM.

In

analogy with the two-dimensional problem (soe tg,

10]),

it is easy to show that

associated withproblem

(1.9)-(1.11)is

given by:

,:,

where(,i,ill)ireell>slmlctionfor thehai, equation

subjecttothemixedboundaryconditions

(1.10)

or

(1.11)

and theinitial condition

limoG

,

z;t) $(

-

),

(3.3)

where $(

-

z)isthe Dirac delta function located at the sourcepoint,

.

Letuswrite

G(

,

;t)=

G0

,

;t)+

x

,

;t),

(3.4)

where

Go(l,2;t)

(4rt)-3/2ep{

₄

(3.5)

is the "fundamentalsolution" of the heat equation

(3.2)

whilex(1,;t)isthe "regularsolution"

chosensothat_G(

,

;t)satisfies the mixedboundary conditions

(1.10)

or

(1.11).

Onsetting wefindthat

O(t)

(4rt)3/2

+K(t),

(3.6)

where

In

what follows, weshalluse Laplacetransforms with respect tot, and use astheLaplace transform parameter; thuswedefine ₊0

An

application of the Laplace transform to the heat equation

(3.2)

shows that

i

,,2;s

2)

satisfiesthe membraue equation

A

s-s)(

t,L

2;s2)

-/i-2)in

,

(3.9)

together withthemixedboundaryconditions

(1.10)

or

(1.11).

Theymptoticexpsion ofK(t)s t0,maythen deduced directly from theymptotic

expsion of

K"

(s

z)

s--, where

4.

CONSTRUCTION

OF

GREEN’S FUNCTION.

It

iswellknown

[7]

thatthe membrane equation

(3.9)

h thefundeutalsolution

0

z, ;

sz)

=e=p(-s

z)

(4.1)

4r

i

where

,

r,,2=

It-21

is the distance between the points

t=,2,)

and

22,

])

of the domain ft. The existence of the solution

(4.1)

enables us to construct integral equations for

,;s

z)

satisfying the mixed boundary conditions

(1.10)

or

(1.11).

Therefore,in Case1,Green’stheorem gives:

i=1Si

dy.

(4.2)

On applying theiterationmethod

(see.

[7], [9], [11])

tothe integralequation (4.2), the

Green’s

function

(

_1,2;s

)

whichhs theregularpart:

{&

,,

z;s

z)

_".

i=1 ~IY it

L

r

it’’-

d

DIMENSIONS 489

where

M,(,V’)=

Z.

K!V}(,V),

(4.4)

v-0

M’(,’)

*K!)(y

",y

),

(4.5)

’=0

L,(

,v

")=

/t’{(

",

),

(4.6)

v----0

L’(y

,y’)=

"K(..v{(V

",,.V.,y

),

(4.7)

0

p(

Sty

(4.8)

Ki(y",y 0

and

emp(

Sry

_Y

")

(4.10)

K_,(y’,)=

r

(4.11)

In

thesameway,wecanshow thatinCase2. the Grin’s function_{G" (}

,

2;s2) h,asaregula,

part of thesameform

(4.3)

withthe interchangesS,,, .S,,i

:

+

m.

On

thebasisof

(4.3)

the function

(

,

and _{2 lie in} the neighborhood of the bounding surfaces Si, i= m of t2 is particulari> interesting. Forthiscase, weneedtousethefollowingcoordinates.

5.

COORDINATES IN THE NEIGHBORHOOD

OF

Si,

m.

Let

h >O(i m) besufficientlysmall.

Let

ni(i m) be theminimumdistances froma

point

=(,z2,za)

of the domain f to its bounding surfaces Si(i= m) respectively. Let

n

i(tri)(i= m) denote theinward drawn unit normals to S,(i m) respectively. We note that the coordinates in the neighborhood of Si(i k

+

m) are in thesameform as in Section

5.1of

[11]

with the interchanges

a

ai,

tr

,

(i k+l m). Thus wehave the sameformulae

(5.1.1)-(5.1.fi)

of Section 5.1 in

[11]

with

the interchanges n a_i,

(a2)

_i(ai),

II!

_IIii,

II2

_II2i,

H

H,

and

N

N,

(i=+

Similarly, the coordinates inthe neighborhoodof Si,(i :)are similar to thoseobtained

0"21

t?,

111 tli, h hi,

I1

li, *(II)

in Section5.2 of

[11]

withthe interchanges

(li) and

6

6i, (i k). Thus, we have the sameformulae

(5.2.1)-(5.2.5)

of Section 5.1 in

[11]

with theinterchanges

n

n,,

n

(%)

N,,

(i k+l, .m)

6.

SOME

LOCAL

EXPANSIONS.

Itnowfollows that the localexpansionsof thefunctions

when the distance between and y is small are very similar tothose obtained in Section 6 of

[11].

Consequently,the local behavior ofthekernels

’,(y

",

),’’_,( ",y

),

(6.2)

g,(g

",g

),

’_,(g"

g

),

(6.3)

when the distance between y and y’is small, follows directly from thelocal expansions of the

functions

(6.

).

DEFINITION

I. If and arepointsin thehalf-part

f3

>0, thenwedefine

An

ex(

x,

z;s)-function

isdefinedfor points and belongtosufficientlysmall domains except when

=

2 li,(i= m) and

,

is called the degree of this function.

For

every

positive integer A,ithas thelocal expansion

(see

[11])"

where

*

denotes asumofafinite number of

erms

in which

J(,)

areinfinitely differentiable functions.

In

this expansion

P, P,

,

,

are integers, where

P

>0,

P

>_0, >0, >0,

,

nB(P

/

P-),

-

4-

/ and the minimum is taken over all terms which occur in the

summationE’. The remainder

RA(

1,

₂) hascontinuousderivatives of orderd<Asatisfying

LtRA(

1,

2;s)=0

[,-

^etp(-

As12) as

whereA isapositive constant.

Thus, using methods similar to those obtained in Section 7 of

[11],

we can show that the

functions

(6.1)

are

e-functions

withdegrees

,

1, 2respectively. Consequently.thefunction.

(6.2)

areeX-functions withdegrees

,

0,- whilethefunctions

(6.3)

aret-functionswith

,

0,1respectively.

DEFINITION

2. If and arepointsinlargedomainsfl

+ S,,

thenwedefine

’,:

mn(r

t

+rt

:

)if

.$i,i=l and

An

Ex

,

;)-function is defined and infinitely differentiable with respect to and whenthesepointsbelong tolargedomainsfl

+

$i exceptwhen ₂ Si, m. Thus, the

r-ftmctioa

haasimilarlocalexpansionof the

-fuaction

(see

[7],

[11]).

With the help of Section 8 in

[11],

it iseasily seen that formula

(4.3)

is an

E-

1, ;)-functionandconsequently

/= m

i=1 i=/+1

which is valid for s--.oo, where Ai(i m) are positive constants. Formula

(6.4)

shows that

0

,

2,s)

is exponentially small for

Withreference to Sections7and 9in

[11],

ifthe -expansions ofthefunctions

(6.1)-(6.3)

are

introduced into

(4.3)

and ifwe useformulae similar to

(7.4)

and

(7.10)

of Section 7 in

[11],

we

DIMENSIONS

(6.5)

where,if

,

and belongtosufficientlysmalldomains(1,), m,then

(

,,

;,2)

_P(-2)+o

as-.

(6.6)

When ,2 >ti, >0,i kand

h,2

> $, >0.i k+1, m, thefunction2(t,2; is of order

0{ep(-sNo) s,N₀>0. Thus,since lira

’

$im

2

(s

[11]),

then the locM

r12__O

R12OP12

havior of the formula

(4.3)

h the form

(6.5),

where if and belong

o

large domains

S,, t, weget

8.12

+

Ot

r12 J

(6.7)

while, if and belongtolargedomainsf

+ S,,

t

+

m,weget:

7.

CONSTRUCTION

OF

RESULTS.

Sincefor

3

>_

hi

>0, m thefunctions ],(

,

;s

2)

are of orders

O(e-

2A,,,,),

the integral

over 1of thefunction ( ,-s

)

canbeapproximatedin the followingway (see

(3.10))"

h!

"

(s2)=

E

,( ,z

"s2){

l-2311,

+({3)2N,}d{3dS,

i=k+l _S,

3=

0 h

E

/ /

,(.t..

’:s2){l+2{311,+({3)2N,}d{3dS,

i=

ls,

3

=0

rtt

2Atsht

+

0( ass-o. (7

i=1

If the

eX-expansions

of ,(

,

;s

2)

are introduced into

(7.1)

and with the help of formula

(10.2)

of Section 10in

[111

wededuce after invertingLaplacetransforms, that

where

and

a a

K(t)

=-T+77-

_+a3

+a4t/2

+O(t)ast-,O,

k

E

[Si[ ’a2=

E

HidS"

k

+

tgX3"2 1"-" _C,

a3 _1-’

S,

E

H2 N,)dS,

i=k+l _S,

"= S, i=k+l

(7.2)

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1.

GOTTLIEB, H.P.W.,

Eigenvaluesofthe Laplacian with Neumann boundary conditions,

J

Austral. Math.Soc. Ser.

B (1985),

293-309.

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HSU, P.,

OntheO-functionofacompact Ricmannian manifold withboundary.

C.R.

Acad. Sci. Set.

I,

309,Paris

(1989),

507-510.

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JR.,

H.P.

&

SINGER, I.M., Curvature

and theeigenvaluesof the Laplacian, J_

Diff.

Geom.

1,

(1967),

43-69.

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PLEIJEL,

A.,

On

Green’s

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Mat.

Konger, XII (1954),

222-240.

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WAECHTER,

R.T.,

Onhearing theshapeofadrum:

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E.M.E.,

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An

extension to higher dimensions,

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(1984),

83-99.

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ZAYED, E.M.E.,

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extension tohigherdimensions,

J.

Math. Anal.Appl. 112

(1985),

455-470.

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ZAYED,

E.M.E.,

Eigenvalues of the Laplacian for the third boundaryvalue problem:

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Math.Anal.Appl. 130

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ZAYED, E.M.E.,

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inverse eigenvalue problem for an arbitrary multiply connected bounded regionin R

2,

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Math. Math. Sci. 14

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ZAYED, E.M.E.,

Hearing theshapeofageneral doubly-connecteddomain in R3_{with mixed}

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